The Stacks project

Lemma 28.20.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. The following are equivalent:

  1. $\mathcal{F}$ is a flat $\mathcal{O}_ X$-module of finite presentation,

  2. $\mathcal{F}$ is $\mathcal{O}_ X$-module of finite presentation and for all $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module,

  3. $\mathcal{F}$ is a locally free, finite type $\mathcal{O}_ X$-module,

  4. $\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module, and

  5. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type, for every $x \in X$ the stalk $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module, and the function

    \[ \rho _\mathcal {F} : X \to \mathbf{Z}, \quad x \longmapsto \dim _{\kappa (x)} \mathcal{F}_ x \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]

    is locally constant in the Zariski topology on $X$.

Proof. This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 10.78.2. The translation uses Lemmas 28.16.1, 28.16.2, 28.19.1, and 28.20.1. $\square$

Comments (2)

Comment #4607 by James Waldron on

Unless this appears elsewhere, it might be useful to add that if is reduced then one can weaken statement (5) to ' is finite type and is locally constant'. i.e. the requirement that is free can be dropped.

(This appears as Exercise 13.7.K in the 17/11/2017 version of Vakil's notes The Rising Sea or Exercise 5.8 in Hartshorne.)

The affine case Lemma 00NX could also include the corresponding statement for finite type modules over reduced rings.

Comment #4777 by on

Thanks! I have added this as a separate lemma in both the algebra and schemes case. See changes.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05P2. Beware of the difference between the letter 'O' and the digit '0'.